568 Lord Rayleigh on the Calculation of the Frequency of 



by 9=—±7r } 6 = -t-^tt. Between the above limits of 6 and 

 when r = c the motion mnst be exclusively tangential. 



In the gravest mode of vibration the fluid swings from one 

 side to the other in such a manner that the horizontal motions 

 are equal and the vertical motions opposite at any two points 

 which are images of one another in the line = 0. This re- 

 lation, which holds also at the two halves of the free surface, 

 implies a stream-function ty which is symmetrical with respect 

 to = 0. 



Let t], denoting the elevation of the surface at a distance 

 r from the centre on the side for which 6 = ^ir, be expressed 



by 



V= -2g i (r/c)+4:q i (r/cf-6q $ (r/ef + . . . ; (4) 



then the potential energy for the whole mass (supposed to be 

 of unit density) is given by 



V=2 f h9V 1 dr=4ge(ltf-iq 2 q i +$q i *+ . . . 



). . (5) 



The more difficult part of the problem lies in determining 

 the motion and in the calculation of the kinetic energy. It 

 may be solved by the method of Sir G. Stokes, who treated a 

 particular case, corresponding in fact to oar first approxima- 

 tion in which (4) reduces to its first term. It is required to 

 find the motion of an incompressible fluid in two dimensions 

 within the semicylinder, the normal velocity being zero o^er 

 the whole of the curved boundary (?* = c, ^7r ->#:> — ^77) and 

 over the flat boundary having values prescribed by (4). If 

 yjr be the stream-function, satisfying d'^/dx" 2 + d 2 \jr/dy 2 =0 1 

 the conditions are that ^ shall be symmetrical with respect 

 to 6 = } that it be constant when r = c from 6 = to 6 = \tt } 

 and that when 6 = ^tt } 



df/dr=d V /dt = -2q 2 {r/c) +4^ 4 (r/c) 3 ~. . ., 

 Or f/c = -U^) 2 + q i (r/c) i -q 6 {r/c) 6 + .., . (6) 



At the edge, where r=c, 



■+/«— -?2 + ?4-?6- ... , .... (7) 



and this value must obtain also over the curved boundary. 

 The conditions may be satisfied* by assuming 



yjr/c = q 2 {r/c) 2 cos 26 + q i (r/c) 4 cos A0 + . . . 



+ 2A 2n+1 (r/c) 2 » +1 cos [2n+l)6, . . (8) 



* Lamb's ( Hydrodynamics,' § 72. 



